Till Lucy has been with you for much, much longer I would not even try treating them while all together. I'm sorry but I definitely would NOT do what you did last night. To me it is unfair to feed the newcomer first, as I said above. It is unfair to taunt the dog by having her near the cat's food and unfair to the cats to have them threatened by the dog. It is unfair to correct the dog because at this point I don't think you can be sure the dog knows the order of things and this is a harsh way to treat her. I'm not against a well place positive correction but in this case what I'd use is a negative punishment of simply being withheld from the room while the cats eat. It's the same difference to you time wise.

Also consider this. Feeding cats first, Lucy removed, then feeding Lucy is a clear indication and positive reward to Lucy for allowing the cats to eat first. CAts finished, YAY, now Lucy gets her reward, her supper. Low key though the YAY is only in your mind, no excitable exclamations.

The reason I fed Lucy first is because I have read that she is less likely to become aggressive over food if she is not hungry. The cats are not in the same room as Lucy when I feed her so they do not know that they are being fed "second". I do agree that it may seem unfair to the cats to be subject to having the dog in the room when they eat but they will learn to be ok with it - just as they have learned to be ok with having a new dog in the house.